Average Power of Waves on Strings: Study with Video Lessons, Practice Problems & Examples

Waves on strings carry energy, not matter, as they propagate through space. To continuously produce waves, average power must be calculated using the equation: ${\frac{1}{2}}^P={\omega }^2{A}^{2}v\mu$. Here, ω is angular frequency, A is amplitude, v is wave speed, and μ is mass density. Understanding these concepts is crucial for solving wave-related problems effectively.

Energy & Power of Waves on Strings

Hey, everyone. So in some problems throughout this chapter, you have to calculate something called the average power. The average power that you have to supply to the string in order to continuously produce waves. I'm going to show you how to do that in this video and it really comes down to just one simple equation that you need to know for the average power of waves on strings. Now before we go any further, I want to point out something really important that you need to know with conceptual point, which is that waves actually carry energy throughout space, not matter. When you whip a string up and down, nothing's actually moving left or to the right. The particles on the string, remember, are actually just going up and down like this. What is happening though is that the overall wave pattern moves to the right and that's because you whip the string up and down. To do that, you had to supply or you had to do some work onto the string. So waves actually carry energy as they go from left to right, not matter. Now in order to continuously produce a wave, you have to continuously flick the string up and down. You have to supply some energy over time. And remember that energy over time is the definition of power. Now in this kind of problems here, you're going to be asked for a specific kind of power, which is the average power, and I'm just going to give you the equation here. Your textbooks are going to do some really lengthy derivation using a bunch of math that you don't need to know. I'm just going to give you the equation because you're most likely going to be given it on our formula sheet. It's 1/2 ω2a2vμ So unfortunately, there's 4 variables in here, but the way I always like to remember it is that this equation kind of looks like the word wave except the mu replaces the e, and also the first two variables are squared. Alright? So it kind of looks like the word wave. Alright. So that's basically it. That's the equation. Let's go ahead and take a look at our example. So we are continuously producing a wave on a taut string. We're told what the mass density is. That's mu, 0.05. We're told that the tension is 100. That's f t is 100. We want to create waves with a frequency of 60. So that's going to be f, and the amplitude of the waves is going to be 6 centimeters. So we have to convert that. That's just going to be a, and that's going to be 0.06. So given these values here, what is the average power? We're just going to use the equation that we just saw up here. I'm going to write it out again. 12ω2a2vμ, so both the first two variables are squared, and then we're going to do v times mu. So it kind of looks like the word wave again. Alright. So which variables do we know? We don't know the angular frequency. We are given the amplitude. We don't know the wave speed, but we are given the mass density. We go out 2 out of 4. I need to find out the other 2 before I plug anything in. So I want to figure out the omega first. So how do we figure out omega? Well, omega unfortunately pops up in a lot of different equations but the one that you should always try out first is this equation here, 2π over period or 2π times frequency. Now hopefully, you realize guys that we're just going to use ω=2πf because we're given here that f is equal to 60. So we just go ahead and do that real quick. This is going to be 2π times 60 and you're going to get omega here of 377 radians per second. Alright. So that's that's done. So now all we have to do is figure out the wave speed. So we have to figure out v. Now remember that we're working a wave we're producing a wave on a string. So we actually have a couple of different equations that we can use. For strings only, we can use this equation here, this v=F_tμ. We can also use Lambda f. Remember because that applies to all kinds of waves. Now if we look through our variables here, we actually have what the tension is and we have what the mass density is. So we're just going to go ahead and stop there and we're just going to use this one right here. So we're going to use, v=100μ which is going to be 0.05. If you go ahead and work this out, what you're going to get is a wave speed of 44.7 meters per second, and it's not squared. Right? So now all we have to do is just plug our values for v and omega back into our equation here. So p average is just going to be 1/2. Now you just plug in everything carefully. We got 377 squared, then we've got our amplitude which is going to be 0.06 squared, And then we've got our V which is 44.7, and then we've got our mass density which is 0.05. If you go ahead and plug in everything carefully, you should get an average power of 572 watts. So that's how much power it takes to continuously produce waves of these characteristics here. Alright. So that's it for this one guys. Let me know if you have any questions.

A horizontal string is stretched with a tension of 90 N, and the speed of transverse waves for the wire is 400 m/s. What must the amplitude of a 70.0 Hz traveling wave be for the average power carried by the wave to be 0.365 W?

• A. 2.9 mm
• B. 4.1 mm
• C. 0.2 mm
• D. 0.017 mm

Here’s what students ask on this topic:

What is the equation for calculating the average power of waves on strings?

The equation for calculating the average power of waves on strings is:

$\frac{1}{2}\pi {\omega }^2{A}^{2}v\mu$

Here, $\omega$ is the angular frequency, $A$ is the amplitude, $v$ is the wave speed, and $\mu$ is the mass density of the string.

How do you calculate the angular frequency for waves on a string?

The angular frequency $\omega$ for waves on a string can be calculated using the equation:

$\omega =2\pi f$

where $f$ is the frequency of the wave. This equation is derived from the relationship between the period and frequency of a wave.

What is the significance of mass density in the average power equation for waves on strings?

Mass density $\mu$ is a crucial factor in the average power equation for waves on strings. It represents the mass per unit length of the string. Higher mass density means the string is heavier, which affects the wave speed and the amount of power needed to sustain the wave. The average power equation is:

$\frac{1}{2}\pi {\omega }^2{A}^{2}v\mu$

where $\mu$ directly influences the power required to produce the wave.

How do you determine the wave speed on a string?

The wave speed $v$ on a string can be determined using the equation:

$v=\sqrt{\frac{T}{\mu }}$

where $T$ is the tension in the string and $\mu$ is the mass density of the string. This equation shows that the wave speed increases with higher tension and decreases with higher mass density.

What is the role of amplitude in the average power of waves on strings?

The amplitude $A$ plays a significant role in the average power of waves on strings. It represents the maximum displacement of the string from its equilibrium position. In the average power equation:

$\frac{1}{2}\pi {\omega }^2{A}^{2}v\mu$

the amplitude is squared, indicating that even small increases in amplitude result in significant increases in the average power required to sustain the wave.